Galois descent on the Brauer group of varieties (joint work with A. N. Skorobogatov)

For a smooth and projective variety $X$ over a field k of characteristic zero we prove the finiteness of the cokernel of the natural map from the Brauer group of $X$ to the Galois-invariant subgroup of the Brauer group of the same variety over an algebraic closure of $k$. Under further conditions on $k$, e.g. over number fields, we give estimates for the order of this cokernel. We emphasise the role played by the exponent of the discriminant groups of the intersection pairing between the groups of divisors and curves modulo numerical equivalence.